The home side stormed out of the blocks and Mike Duff headed them into the lead from a Wade Elliott free-kick.
Paul McVeigh held off a defender to guide in a second after keeper Brian Jensen's goal-kick had been flicked on.
Steve Jones lashed in a shot and Elliott fired low inside the far post to complete a win before Sylvain Ebanks-Blake missed a Plymouth penalty.
Burnley manager Steve Cotterill:
"I'm delighted with it, delighted with the performance, effort and application of the players. They were first class.
"I'm delighted for the chairman and the directors, who have stood by me, and the supporters, who have stood by me, because it's been tough.
"Early mornings, late nights, waking up in the night and writing things down so I don't forget them, trying them the next morning.
"The time that's gone on has been the toughest I've had to face in my managerial career."
Plymouth manager Ian Holloway:
"It's the worst performance in my football career.
"It's an absolute disgrace. That was embarrassing. It looked like we were playing a testimonial game and they were playing a real match.
"It's unacceptable and I apologise wholeheartedly to the fans. I suggest that the players pay for their tickets and their travel."
Burnley: Jensen, Thomas, Duff, Caldwell, Harley, Elliott (Garreth O'Connor 76), James O'Connor (McCann 70), Djemba-Djemba, Jones, Gray (Akinbiyi 85), McVeigh.
Subs Not Used: Coyne, Coughlan.
Goals: Duff 13, McVeigh 20, Jones 38, Elliott 61.
Plymouth: McCormick, Connolly, Seip, Timar, Capaldi, Gosling (Djordjic 72), Hodges (Buzsaky 35), Nalis, Sinclair, Hayles, Fallon (Ebanks-Blake 60).
Subs Not Used: Halmosi, Sawyer.
Booked: Fallon, Connolly, Seip, Ebanks-Blake.
Ref: A Hall (W Midlands).